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Full text of "Treatise On Analysis Vol-Ii"

346        XV    NORMED ALGEBRAS AND SPECTRAL THEORY

Let s\~*U(s) be a representation of A in a Hilbert space H. A vector
subspace E of H is said to be stable with respect to this representation if
U(s)E c E for all s e A. If E is stable with respect to /, then so is the
closure E of E (3.11.4). If E is closed and if E' is the orthogonal supplement
of E in H (6.3), then E' is also stable with respect to U. For if x e E and
x' E E', we have (x \ U(s)  x') = ((U(s))*x \ x') = (U(s*)x | x') = 0 by hypo-
thesis, hence U(s) - x' is orthogonal to all x e E and therefore belongs to E'.
If E/iCs1) and U2(s) are the restrictions of U(s) to E and E', respectively, then
the representation U is the Hilbert sum of Ut and U2 -

(15.5.3) For a closed subspace EofH to be stable with respect to U, it is
necessary and sufficient that PE U(s) = U(s)PE for all seA, where PE is the
orthogonal projection on E (6.3).

The condition is necessary, for if x e E we have PE - x = x and
PE * (U(s) - x) = U(s) - x, because E is stable with respect to U; and if
x e E' we have PE - x = 0 and PE  (U(s) * x) = 0, because E' is also stable
with respect to U. Conversely, if the given condition is satisfied, then
U(s) - x = PE - U(s) - x e E for all x e E and all s e A.

On a Hilbert space H, an orthogonal projector is by definition any con-
tinuous operator on H which is an orthogonal projection onto a closed sub-
space of H. The importance of such projectors is due to (15.5.3) and to their
characterization in terms of the structure of algebra with involution of
JSP(H):

(15.5.3.1) A continuous operator P on a Hilbert space H is an orthogonal
projector if and only if it is idempotent and hermit tan.

The necessity of these conditions has already been proved (11.5). Con-
versely, if P2 = P = P*, then (P  x\y - P ' y) = (x\P - y - P2  y) = 0 for
all x, y e H. Since P(H) is also the kernel of 1H  P, it is closed, and H is
the Hilbert sum of P(H) and P ~'(()). Hence the result.

Suppose that H is the Hilbert sum of an infinite sequence (Hn) of sub-
spaces which are stable with respect to the representation U. Let Un(s) denote
the restriction of U(s) to Hn, so that for each n the mapping s\-* Un(s) is a
representation of A in Hn. By abuse of language, tlie representation U is said
to be the Hilbert sum of the representations Un. For each s e A and each
x = Z xn6 H, where xn e Hn for each , we have U(s)  x =  Un(s) - xn, and